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dynare/matlab/th_autocovariances.m

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function [Gamma_y,ivar]=th_autocovariances(dr,ivar)
% function [Gamma_y,ivar]=th_autocovariances(dr,ivar)
% computes the theoretical auto-covariances, Gamma_y, for an AR(p) process
% with coefficients dr.ghx and dr.ghu and shock variances Sigma_e_
% for a subset of variables ivar (indices in lgy_)
%
% INPUTS
% dr: structure of decisions rules for stochastic simulations
% ivar: subset of variables
%
% OUTPUTS
% Gamma_y: theoritical auto-covariances
% ivar: subset of variables
%
% SPECIAL REQUIREMENTS
% Theoretical HP filtering is available as an option
%
% part of DYNARE, copyright Dynare Team (2001-2008)
% Gnu Public License.
global M_ options_
exo_names_orig_ord = M_.exo_names_orig_ord;
if sscanf(version('-release'),'%d') < 13
warning off
else
eval('warning off MATLAB:dividebyzero')
end
nar = options_.ar;
Gamma_y = cell(nar+1,1);
if isempty(ivar)
ivar = [1:M_.endo_nbr]';
end
nvar = size(ivar,1);
ghx = dr.ghx;
ghu = dr.ghu;
npred = dr.npred;
nstatic = dr.nstatic;
kstate = dr.kstate;
order = dr.order_var;
iv(order) = [1:length(order)];
nx = size(ghx,2);
ikx = [nstatic+1:nstatic+npred];
k0 = kstate(find(kstate(:,2) <= M_.maximum_lag+1),:);
i0 = find(k0(:,2) == M_.maximum_lag+1);
i00 = i0;
n0 = length(i0);
AS = ghx(:,i0);
ghu1 = zeros(nx,M_.exo_nbr);
ghu1(i0,:) = ghu(ikx,:);
for i=M_.maximum_lag:-1:2
i1 = find(k0(:,2) == i);
n1 = size(i1,1);
j1 = zeros(n1,1);
for k1 = 1:n1
j1(k1) = find(k0(i00,1)==k0(i1(k1),1));
end
AS(:,j1) = AS(:,j1)+ghx(:,i1);
i0 = i1;
end
b = ghu1*M_.Sigma_e*ghu1';
ipred = nstatic+(1:npred)';
% state space representation for state variables only
[A,B] = kalman_transition_matrix(dr,ipred,1:nx,dr.transition_auxiliary_variables);
if options_.order == 2 | options_.hp_filter == 0
[vx, u] = lyapunov_symm(A,B*M_.Sigma_e*B');
iky = iv(ivar);
if ~isempty(u)
iky = iky(find(all(abs(ghx(iky,:)*u) < options_.Schur_vec_tol,2)));
ivar = dr.order_var(iky);
end
aa = ghx(iky,:);
bb = ghu(iky,:);
if options_.order == 2 % mean correction for 2nd order
Ex = (dr.ghs2(ikx)+dr.ghxx(ikx,:)*vx(:)+dr.ghuu(ikx,:)*M_.Sigma_e(:))/2;
Ex = (eye(n0)-AS(ikx,:))\Ex;
Gamma_y{nar+3} = AS(iky,:)*Ex+(dr.ghs2(iky)+dr.ghxx(iky,:)*vx(:)+...
dr.ghuu(iky,:)*M_.Sigma_e(:))/2;
end
end
if options_.hp_filter == 0
Gamma_y{1} = aa*vx*aa'+ bb*M_.Sigma_e*bb';
k = find(abs(Gamma_y{1}) < 1e-12);
Gamma_y{1}(k) = 0;
% autocorrelations
if nar > 0
vxy = (A*vx*aa'+ghu1*M_.Sigma_e*bb');
sy = sqrt(diag(Gamma_y{1}));
sy = sy *sy';
Gamma_y{2} = aa*vxy./sy;
for i=2:nar
vxy = A*vxy;
Gamma_y{i+1} = aa*vxy./sy;
end
end
% variance decomposition
if M_.exo_nbr > 1
Gamma_y{nar+2} = zeros(length(ivar),M_.exo_nbr);
SS(exo_names_orig_ord,exo_names_orig_ord)=M_.Sigma_e+1e-14*eye(M_.exo_nbr);
cs = chol(SS)';
b1(:,exo_names_orig_ord) = ghu1;
b1 = b1*cs;
b2(:,exo_names_orig_ord) = ghu(iky,:);
b2 = b2*cs;
vx = lyapunov_symm(A,b1*b1');
vv = diag(aa*vx*aa'+b2*b2');
for i=1:M_.exo_nbr
vx1 = lyapunov_symm(A,b1(:,i)*b1(:,i)');
Gamma_y{nar+2}(:,i) = abs(diag(aa*vx1*aa'+b2(:,i)*b2(:,i)'))./vv;
end
end
else
if options_.order < 2
iky = iv(ivar);
aa = ghx(iky,:);
bb = ghu(iky,:);
end
lambda = options_.hp_filter;
ngrid = options_.hp_ngrid;
freqs = 0 : ((2*pi)/ngrid) : (2*pi*(1 - .5/ngrid));
tpos = exp( sqrt(-1)*freqs);
tneg = exp(-sqrt(-1)*freqs);
hp1 = 4*lambda*(1 - cos(freqs)).^2 ./ (1 + 4*lambda*(1 - cos(freqs)).^2);
mathp_col = [];
IA = eye(size(A,1));
IE = eye(M_.exo_nbr);
for ig = 1:ngrid
f_omega =(1/(2*pi))*( [inv(IA-A*tneg(ig))*ghu1;IE]...
*M_.Sigma_e*[ghu1'*inv(IA-A'*tpos(ig)) ...
IE]); % state variables
g_omega = [aa*tneg(ig) bb]*f_omega*[aa'*tpos(ig); bb']; % selected variables
f_hp = hp1(ig)^2*g_omega; % spectral density of selected filtered series
mathp_col = [mathp_col ; (f_hp(:))']; % store as matrix row
% for ifft
end;
% covariance of filtered series
imathp_col = real(ifft(mathp_col))*(2*pi);
Gamma_y{1} = reshape(imathp_col(1,:),nvar,nvar);
% autocorrelations
if nar > 0
sy = sqrt(diag(Gamma_y{1}));
sy = sy *sy';
for i=1:nar
Gamma_y{i+1} = reshape(imathp_col(i+1,:),nvar,nvar)./sy;
end
end
%variance decomposition
if M_.exo_nbr > 1
Gamma_y{nar+2} = zeros(nvar,M_.exo_nbr);
SS(exo_names_orig_ord,exo_names_orig_ord)=M_.Sigma_e+1e-14*eye(M_.exo_nbr);
cs = chol(SS)';
SS = cs*cs';
b1(:,exo_names_orig_ord) = ghu1;
b2(:,exo_names_orig_ord) = ghu(iky,:);
mathp_col = [];
IA = eye(size(A,1));
IE = eye(M_.exo_nbr);
for ig = 1:ngrid
f_omega =(1/(2*pi))*( [inv(IA-A*tneg(ig))*b1;IE]...
*SS*[b1'*inv(IA-A'*tpos(ig)) ...
IE]); % state variables
g_omega = [aa*tneg(ig) b2]*f_omega*[aa'*tpos(ig); b2']; % selected variables
f_hp = hp1(ig)^2*g_omega; % spectral density of selected filtered series
mathp_col = [mathp_col ; (f_hp(:))']; % store as matrix row
% for ifft
end;
imathp_col = real(ifft(mathp_col))*(2*pi);
vv = diag(reshape(imathp_col(1,:),nvar,nvar));
for i=1:M_.exo_nbr
mathp_col = [];
SSi = cs(:,i)*cs(:,i)';
for ig = 1:ngrid
f_omega =(1/(2*pi))*( [inv(IA-A*tneg(ig))*b1;IE]...
*SSi*[b1'*inv(IA-A'*tpos(ig)) ...
IE]); % state variables
g_omega = [aa*tneg(ig) b2]*f_omega*[aa'*tpos(ig); b2']; % selected variables
f_hp = hp1(ig)^2*g_omega; % spectral density of selected filtered series
mathp_col = [mathp_col ; (f_hp(:))']; % store as matrix row
% for ifft
end;
imathp_col = real(ifft(mathp_col))*(2*pi);
Gamma_y{nar+2}(:,i) = abs(diag(reshape(imathp_col(1,:),nvar,nvar)))./vv;
end
end
end
if sscanf(version('-release'),'%d') < 13
warning on
else
eval('warning on MATLAB:dividebyzero')
end
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